Affine semigroup conjecture for the f-vector set of 4-polytopes under connected sum addition

Let $\mathcal{F}(\mathcal{P}^4)$ denote the set of $f$-vectors of all $4$-polytopes, and let $\mathbin{{\scriptstyle\boxplus}”}$ be the modified addition realized by connected sum operations. Affine semigroup conjecture. The $f$-vector set of all $4$-polytopes is closed under the modified addition $\mathbin{{\scriptstyle\boxplus}”}$. That is, $\big(\mathcal{F}(\mathcal{P}^4),\mathbin{{\scriptstyle\boxplus}”}\big)$ is an affine semigroup. The preceding proposition establishes the analogous statement for $3$-polytopes, while the claim for $4$-polytopes is left as a conjecture; the source gives no resolution.