Cohomological lower-bound conjecture for distributional sectional category

Let $p:E\to B$ be a map, let $n\ge 1$, and let

$$p_n:SP^{n!}(E)\to SP^{n!}(B)$$

be induced by the functor $SP^{n!}$. Let

$$\delta_n:B\to SP^{n!}(B)$$

be the diagonal inclusion. For a ring $R$, suppose that $\alpha_i^*\in H^{k_i}(SP^{n!}(B);R)$, $1\le i\le n$, with $k_i\ge 1$, satisfy $p_n^*(\alpha_i^*)=0$. Let $\alpha_i=\delta_n^*(\alpha_i^*)$.

Proposed cohomological lower bound. If

$$\alpha_1\smile\cdots\smile\alpha_n\ne 0,$$

then

$$\operatorname{\mathsf{dsecat}}(p)\ge n.$$

This is proposed as a generalization of Schwarz's cohomological lower bound for sectional category and of an earlier lower-bound result for distributional topological complexity. The supplied text does not state whether the proposed bound has been proved or disproved.