The combinatorial Hodge conjecture for projective toric varieties

Let $\Sigma$ be a rational fan defining a projective toric variety $X_\Sigma$. The combinatorial intersection cohomology is denoted $IH_{\mathrm{comb}}^*(\Sigma,\mathbb{Q})$, and for each $k\geq 0$ let $Hdg^k_{\mathrm{comb}}(\Sigma)$ be the span of the combinatorial cycle classes $[V(\tau)]_{\mathrm{comb}}$ associated with cones $\tau\in\Sigma$ of dimension $k$. Let

$$\phi:IH_{\mathrm{comb}}^*(\Sigma,\mathbb{Q})\xrightarrow{\sim}IH^*(X_\Sigma,\mathbb{Q})$$

be the canonical isomorphism. The combinatorial Hodge conjecture. This isomorphism should map the combinatorial Hodge classes precisely onto the geometric Hodge classes:

$$\phi\left(Hdg^k_{\mathrm{comb}}(\Sigma)\right)=\mathrm{Hdg}^k(X_\Sigma).$$

Equivalently, using the source's assertion that $\mathrm{Hdg}^k(X_\Sigma)=IH^{2k}(X_\Sigma,\mathbb{Q})$ for projective toric varieties, the combinatorial cycle classes should span $IH^{2k}_{\mathrm{comb}}(\Sigma,\mathbb{Q})$. The paper reports verification in dimensions at most $3$ and for simplicial projective toric varieties, while the general claim remains open in the supplied text.