The forbidden-configuration bound for binary matrices

Let $t$ be given, and let ${\mathrm{forb}}(m,F(0,2^{t-2}+1,1,0))$ denote the maximum number of columns in an $m$-rowed binary matrix avoiding the configuration $F(0,2^{t-2}+1,1,0)$. Forbidden-configuration bound conjecture.

$${\mathrm{forb}}(m,F(0,2^{t-2}+1,1,0))\leq \frac{2^{t}-1}{t}m+1,$$

with equality when $m\equiv 0\pmod{t}$. The conjecture is motivated by constructions using the matrix $K_t$ and the matrices $B_i=K_t\backslash \mathbf{1}_t$, which attain the stated bound in the displayed cases; its validity for arbitrary $t$ is not established in the supplied text.