Forbidden-configuration product conjecture for simple matrices

Let $F$ be a simple $k\times \ell$ $(0,1)$-matrix. A product of matrices is obtained by stacking, in order, one column from each factor; the factors here are identity matrices $I$, their $(0,1)$-complements $I^c$, and triangular matrices $T$, with row and column permutations ignored. Let $t$ be the largest integer for which there is a $t$-fold product of matrices chosen from $I$, $I^c$, and $T$ that avoids $F$. Product conjecture. Then

$$\operatorname{forb}(m,F)=\Theta(m^t).$$

This conjecture predicts that the asymptotic growth of the extremal number of simple matrices avoiding $F$ is achieved by products of identity, complemented identity, and triangular matrices. The supplied text gives no resolution or further scope for the claim.