Strictness conjecture for constructible-sheaf quantifier levels

For each positive integer $p$, let $\boldsymbol{\Lambda}^{(p)}\boldsymbol{\mathcal{P}}_\mathbb{R}$ denote the $p$-th sheaf-theoretic quantifier level built from $\boldsymbol{\mathcal{P}}_\mathbb{R}$. Constructible-sheaf hierarchy strictness conjecture. For every $p>0$,

$$\boldsymbol{\Lambda}^{(p)}\boldsymbol{\mathcal{P}}_\mathbb{R}\neq\boldsymbol{\mathcal{P}}_\mathbb{R}.$$

The source notes that this is a priori weaker than the standard conjecture separating the corresponding compact real BSS hierarchy levels from $\mathbf{P}_\mathbb{R}^c$.