Three parties have been charged in relation to the fire - the mall's owner, Doha municipality and Gympanzee - but none of them showed up when due in court this month.
On Friday, part of the mall reopened to the public, prompting the parents of the 13 children to write a collective letter hitting out at the owners and management, who they said should be "ashamed", and Mr Weekes to speak out for the first time since the disaster.
"It's just like, 'Let's forget it ever happened', and I think that's another tragedy waiting to happen," Mr Weekes said.
"We will never get our children back ... but we don't want to see any other parent, any other sister, any other brother be put in that situation. Something significant went wrong when at 11 o'clock on a Tuesday, 19 people died in the biggest, most modern mall in the country. How can that happen?"
Doha News reported there was a large security presence within the mall when its doors reopened. Security had been doubled or tripled at every entrance and exit. Management officials were noticeable around the mall.
Posted in every shop window were notices listing safety rules and the site of fire extinguishers.