A question related to the strong Oda conjecture

Question

A fan is a collection of strongly convex rational polyhedral cones in $\mathbb Z^n$, which we often think of as contained in $\mathbb Q^n$ or $\mathbb R^n$ for purposes of visualizing it. The defining property of a fan is that the intersection of any two cones of the fan is a face of each.

A fan is called unimodular if, for each cone of the fan, the first lattice point along each of the rays forms a basis for the lattice in the linear span of the cone. (We call these lattice points the generators of the rays.)

Given a fan $\Sigma$ and a cone $\sigma$ of $\Sigma$, we can subdivide $\Sigma$ at $\sigma$ by introducing a new ray which is generated by the sum of the generators of $\sigma$, and then subdividing $\sigma$ and the cones containing $\sigma$ in a standard way so as to take advantage of the new ray. If $\Sigma$ was unimodular, so is this subdivision of $\Sigma$.

Given a fan, there is a corresponding toric variety. Subdividing at $\sigma$ corresponds to blowing up the toric variety at a the smooth subvariety corresponding to $\sigma$. My question is really just about fans, so there is no need to think about the toric varieties, but I'm still going to use the term (smooth) blow up for the subdivision, and blow down for the reverse process.

Given two unimodular fans which cover the same region, the weak Oda conjecture says that you can turn one into the other by a sequence of (smooth) blowups and blowdowns. This was established independently by Włodarczyk and Morelli. The strong Oda conjecture says that you can first do all the blowups, then all the blowdowns. To the best of my knowledge, it remains open.

My question is the following: Suppose the fan $\Sigma'$ is a refinement of the fan $\Sigma$. Is there a sequence of smooth blowups that gets you from $\Sigma$ to $\Sigma'$, or do you still potentially have to do some blowdowns?

Edited: I initially claimed that my question just amounted to asking whether the Strong Oda conjecture is known to hold for $\Sigma$ and $\Sigma'$ as in the previous paragraph, but I am in fact asking for something stronger, since the Strong Oda conjecture would allow you to go from $\Sigma$ to $\Sigma'$ by a sequence of blowups followed by blowdowns, whereas I want to go via only blowups.

Answer 1

I realized, after asking the question, that if the original question was known to have an affirmative answer, then that would imply the strong Oda conjecture. Since the strong Oda conjecture isn't known to be true, this statement mustn't be known either. (I explain the argument below.)

A negative answer to the original question doesn't imply the falsity of the strong Oda conjecture, though, so it's possible that the answer to the original question is known or can be easily shown to be negative. A reasonable MathOverflow-type question therefore remains to be answered. (Of course, it would also be fine if someone wants to prove the strong Oda conjecture, but this is probably not the place to do it.)

Deducing the strong Oda conjecture from an affirmative answer to the question: Let $\Sigma$ and $\Psi$ be two unimodular fans covering the same region, as in the strong Oda conjecture. Let $\Phi$ be their common refinement. This is a fan, but needn't be unimodular. However, it can be refined to a fan $\Phi'$ which is unimodular. Then, an affirmative answer to the question would imply we could pass from $\Sigma$ to $\Phi'$ by a sequence of blowups, and from $\Phi'$ to $\Psi$ by a sequence of blowdowns, thus proving the strong Oda conjecture.