Sheaf-theoretic Toda conjecture

For a sequence of constructible sheaves in the real sheaf-theoretic polynomial hierarchy $\boldsymbol{\mathcal{PH}}_\mathbb{R}$, let $\mathbf{Eu}(\boldsymbol{\mathcal{PH}}_\mathbb{R})$ be the class of sequences of constructible functions obtained by taking the Euler characteristic of stalk cohomology. Let $\#\boldsymbol{\mathcal{P}}_\mathbb{R}$ be the corresponding counting-type class. Sheaf-theoretic Toda conjecture.

$$\mathbf{Eu}(\boldsymbol{\mathcal{PH}}_\mathbb{R})=\#\boldsymbol{\mathcal{P}}_\mathbb{R}.$$

This is proposed as a sheaf-theoretic reformulation of Toda's theorem, relating the polynomial hierarchy to a counting class; its validity in this setting remains open.