After
losing their way to be all out for 378, New Zealand claimed early wickets to have England reeling at 18-2, but an unbroken third-wicket stand of 93 between the idiosyncratic Rory Burns, 59, and Joe Root, 42, means the hosts trail by 267.
Conway's heroics – he was unlucky not to carry his bat after being sold down the river by Neil Wagner soon after reaching 200 with a pull for six – place him in eternity on the Lord's honours board but it doesn't tell the half of it. His was the second-highest score by a New Zealander on debut, the highest at Lord's and the seventh-highest of all time.
He was an inch from mirroring Charles Bannerman's feat of a century and carrying the bat on debut, achieved in 1877.
Conway's brilliance, however, couldn't mask an awkward truth for New Zealand: two of their all-time greats are struggling big time. Ross Taylor's struggles on day one were matched by BJ Watling on day two.
In normal circumstances New Zealand's modus operandi is to persevere, but these are not normal circumstances.
A conservative selection policy works well within the bilateral test series framework, when you need to build relationships, partnerships and a wellspring of experience, but the big prize in England is the World Test Championship – a one-off, winner-takes-all match. The selectors' role is, simply, to pick the players in the best position to do their roles well in that test.
Picking purely on loyalty or, worse, sentimentality, would be an abdication of their responsibility.
So the conversation needs to be had: Does Will Young or Daryl Mitchell offer a better chance of runs at No 4 than Taylor, and likewise does Tom Blundell make more sense on form than Watling?
You suspect these are conversations coach Gary Stead and Kane Williamson would rather not have due to the pedigree of the players in question. A strong second innings would give everybody involved comfort in this finely poised test.
It's poised because New Zealand could not capitalise on the graft of day one.
At 288-3, with Conway and rock-solid Henry Nicholls eyeing a 200-run partnership, the only question seemed to be how many New Zealand would get before looking to declare.
Before the 300 was brought up England had not only pulled the visitors back, but arguably taken the upper hand.
The bowling from debutant Ollie Robinson and Mark Wood in particular was superb but the batting of a middle order loaded with all-rounders, from keeper Watling through to Kyle Jamieson, was miserable.
Nicholls shovelled a Wood bumper that got big on him to fine leg and just like that, the wheels fell off and started rolling down St John's Wood Rd. New Zealand have not been immune to collapses at the Home of Cricket ™ but this was more inexplicable than most.
Watling, one, tried to work a ball on off stump into the leg side and exposed his leading edge. Dom Sibley snaffled him at second slip. That's seven single-figure scores and just one half century in his last 13 innings as he limps towards the finishing line.
Colin de Grandhomme and Mitchell Santner, picked as much for batting insurance as he is for his left-arm orthodox, failed to register. The former pushed tentatively forward and was trapped on the pads while the latter chipped tamely to a shortish mid-off.
Jamieson, nine, at least saw it through to lunch, but he forgot that he had a guy in decent nick at the other end and instead tried to dominate. He was set up line, sinker and top-edged hook.
New Zealand's strength coming into the match – a middle order that batted to No 9 – turned out to be their nemesis.
Still the immaculate Conway pressed on. He wasn't entirely blameless for the predicament as he became marooned in the 160s and couldn't ease the pressure on the other end, but if you were apportioning liability for the collapse it would be Conway 0.4 per cent, the rest 99.6.
Southee, eight, gave a life then nicked James Anderson behind, which brought Wagner to the crease.
He was spectacular, peeling off punches on the up through the covers, a lofted drive for six off Stuart Broad and eventually seeing Conway through to his double ton before running him out looking for two in the deep.
An innings of high promise ended, remarkably, with Wagner as third-top scorer on 25, with extras next a run further back.
Still, 378 looked very good when England went to tea at 25-2.
Jamieson, taking the new ball in Trent Boult's absence, defeated Sibley leg before for a duck and Southee lured Zak Crawley, two, into a loose shot wide of off and Watling gleefully accepted the nick.
That was to be the last success as Burns and Root dug in for the fight against a probing but slightly one-paced attack.
The first session on day three now shapes as the most pivotal of the match and, perhaps, the series.
