The equivariant blow-up decomposition conjecture for nonsingular toric Fano varieties

Let $X$ and $Y$ be nonsingular toric Fano $d$-folds, and let $\varphi:Y\longrightarrow X$ be an equivariant morphism. An equivariant blow-up is the blow-up along a $T_N$-invariant irreducible closed subvariety. Equivariant blow-up decomposition conjecture. The morphism $\varphi$ should admit a decomposition

$$Y=X_r\stackrel{\varphi_r}{\longrightarrow}X_{r-1}\stackrel{\varphi_{r-1}}{\longrightarrow}\cdots\stackrel{\varphi_2}{\longrightarrow}X_1\stackrel{\varphi_1}{\longrightarrow}X_0=X$$

where each $X_i$ for $0\leq i\leq r$ is a nonsingular toric Fano $d$-fold, and each $\varphi_j$ for $1\leq j\leq r$ is an equivariant blow-up along a $T_N$-invariant irreducible closed subvariety of $X_{j-1}$.

This would extend the three-dimensional decomposition theorem to arbitrary dimension and is presented as a possible continuation of the classification method for $d\geq4$. The source gives no resolution beyond the established three-dimensional case.