Colon ideal conjecture for generalized star configurations

Let $cmathbb K$ be a field of characteristic $0$, let $cmathcal C=(cell_1,\ldots,cell_n)\subset R=cmathbb K[x_1,\ldots,x_k]$ be an $[n,k]$-linear code with no zero columns, and let $I_a(cmathcal C)$ be generated by all $a$-fold products of its linear forms. For $cellincmathcal C$, write $cmathcal C'=cmathcal Csetminuscell$. Colon ideal conjecture. For every $cmathcal Csubset R$, every $cellincmathcal C$, and every $2\leq aleq n$, one has

$$\operatorname{I}_a(cmathcal C):cell=I_{a-1}(cmathcal C').$$

This equality is known for usual star configurations and is conjectured here for arbitrary codes; it is intended as a key step toward understanding their homological properties.