Akutagawa's radiality conjecture for Yamabe constants on products with Euclidean space

Let $(M_1^{m_1},h_1)$ be a Yamabe metric of positive scalar curvature, let $m_2\geq 2$, and equip $M_1\times\mathbb{R}^{m_2}$ with the product metric $h_1+g_{\operatorname{Flat}}$. The Yamabe constant is understood as the infimum over compactly supported functions in $H^1(M_1\times\mathbb{R}^{m_2})$.

Akutagawa's radiality conjecture. This infimum is achieved by a function radial in the $\mathbb{R}^{m_2}$ variable.

The conjecture is used to analyze the limit of Yamabe constants of stretched products and would yield a lower bound for $\sigma(S^2\times S^2)$. The source does not provide a resolution.