Multiplicity-free inverse Grassmannian Schubert product conjecture

Let $\mathfrak S_w$ denote the Schubert polynomial indexed by a permutation $w$. A permutation with a unique left descent is called inverse Grassmannian. For inverse Grassmannian permutations $u$ and $v$, consider the product $\mathfrak S_u\cdot\mathfrak S_v$.

Multiplicity-free inverse Grassmannian conjecture. If $u$ and $v$ are inverse Grassmannian, then $\mathfrak S_u\cdot\mathfrak S_v$ is a multiplicity-free sum of Schubert polynomials.

The statement is equivalent to all Schubert structure coefficients in this product being $0$ or $1$. It was verified for all permutations in $S_9$, but the source gives no proof in general. The analogous assertion for Grassmannian permutations is false in general.