Linear-kernel dimension conjecture for HGP codes based on LPS graphs

Fix $t$, and let $n$ be large enough that the parameters $m_2,\ldots,m_t$ defined by

$$m_1=\lceil 2\sqrt{n-1}\rceil,\qquad m_2=\cdots=m_t=\frac{n}{2}-\frac{2}{t-1}\lfloor\sqrt{n-1}+c\rfloor,$$

where $c\in(-1,1)$ is chosen so that $\sum_i m_i=n+1$. Let $H$ be the Tanner-code parity-check matrix appearing in the construction of the HGP codes based on an LPS graph family. Kernel-dimension conjecture. If $m_2,\ldots,m_t$ are close enough to $n/2$, then

$$\dim\ker H=\Theta(V),$$

so that the resulting code dimension satisfies $k=\Theta(V^3)=\Theta(N)$. This would provide the missing tight lower bound on $k$ while retaining the stated HGP code parameters and nontrivial constant-depth cup-product gates; the claim is presented as a conjecture in the source and no resolution is given.