The 2-blow-up conjecture for totally nondegenerate embeddings

Let $X$ and $\widetilde{X}$ be nonsingular projective toric $4$-folds, and let $\psi:\widetilde{X}\rightarrow X$ be a 2-blow-up, meaning an equivariant blow-up along a $T_N$-invariant subvariety of codimension $2$. 2-blow-up conjecture. If $X$ admits no totally nondegenerate embedding, then $\widetilde{X}$ admits no totally nondegenerate embedding either. This is the embedding analogue of the preceding proposition for totally nondegenerate finite morphisms; the source proposes it as a conjecture, and no resolution is given here.