An Ice Block scheduled for construction lists its available errands as Construction Supply, Building or Supplying.
Exhibit A: Ice Block Errands
Poor Marie has been locked in a carbon dioxide room until she transfers ice from the storage bin to the ice block. Unfortunately it seems she is not going to transfer the ice and instead stands idle.
Exhibit B: Marie's stubbornness
Although Marie has the Unconstructive trait, she is able to do Supply jobs and in fact they are a high priority for her. And since supply jobs includes the errand 'Construction Supply', she should be able to deliver the ice in much the same way that she can make all sorts of other deliveries.
It makes sense that Unconstructive would block errands exclusive to Building jobs (namely Deconstruct and Build), but not errands that are also available as Supply jobs.
Please note: if this issue isn't addressed soon, she may die!