Polynomial rank-bound conjecture for depth-four identities

Let $K$ be a field of characteristic $\operatorname{ch}(K)$, and let $R_{\delta}(k,s)$ strictly upper-bound the rank of any minimal and simple $\Sigma\Pi_{\delta}\Sigma\Pi(k,s,n)$ identity, where the rank is the transcendence degree over $K$ of the set of sparse polynomials in the circuit. The known constructions give lower bounds $R_{\delta}(k,s)=\operatorname{\Omega}(\delta k)$ when $\operatorname{ch}(K)=0$ and $R_{\delta}(k,s)=\operatorname{\Omega}(\delta k\log_p s)$ when $\operatorname{ch}(K)=p>0$. Rank-bound conjecture.

$$R_{\delta}(k,s)= \begin{cases} \operatorname{poly}(\delta k), & \text{if $\operatorname{ch}(K)=0$},\\ \operatorname{poly}(\delta k\log s), & \text{otherwise}. \end{cases}$$

This conjecture would give polynomial upper bounds on the rank parameter governing minimal and simple depth-four identities, a quantity central to deterministic blackbox identity testing. The source does not provide a resolution of the conjecture.