The generalized WAX decomposition validity conjecture

Let ${\boldsymbol{A}}$ be given by the Kronecker construction in the source, with $\widetilde{{\boldsymbol{A}}}_\mathrm{T}=\mathbf{I}_{T_{\mathrm{P}}}$, and let $\widetilde{{\boldsymbol{A}}}_\mathrm{B}$ be constructed through the stated generalized algorithm. For each iteration $i=1,\dots,N_\text{Q}$, let $Q_i$ be the quotient defined by that algorithm, and let $N_\text{Q}$ be the iteration at which the dimensions are exhausted. Define

$$Q_\mathrm{tot}=Q_1+\frac{1}{Q_2+\frac{1}{\ddots+\frac{1}{Q_{N_\text{Q}}}}}.$$

Generalized WAX decomposition validity conjecture. A randomly chosen matrix ${\boldsymbol{H}}$ admits WAX decomposition with probability $1$ for the given ${\boldsymbol{A}}$ if

$$L\geq \frac{K}{1+Q_\mathrm{tot}}.$$

Moreover, in the regime $T_{\mathrm{P}}<M_{\mathrm{P}}/2+1$, the first column of $\widetilde{{\boldsymbol{A}}}_\mathrm{B}$, written as $[\alpha_1,\dots,\alpha_\Phi]^\mathrm{T}$, should satisfy the same restrictions as in Proposition 2 of the source. The conjecture is motivated by the proposed generalization of earlier valid constructions. A formal proof for general $N_\text{Q}$ was not found, although extensive simulations reportedly found no exceptions and produced valid matrices exactly when the stated condition was satisfied.