First-order free group factor alternative

For $r\in(1,\infty]$, let $L(\mathbb F_r)$ be the interpolated free group factor, and let the first-order fundamental group of $L(\mathbb F_r)$ be the group of positive scalars $t$ such that the amplification of $L(\mathbb F_r)$ by $t$ is elementarily equivalent to $L(\mathbb F_r)$. Let $\equiv$ denote elementary equivalence. First-order free group factor alternative. One of the following two statements must hold:

1. $L(\mathbb F_r)\equiv L(\mathbb F_s)$ for all $1<r\leq s\leq\infty$, and the first-order fundamental group of $L(\mathbb F_r)$ is $\mathbb R_+$ for all $1<r\leq\infty$.
2. $L(\mathbb F_r)\not\equiv L(\mathbb F_s)$ for all $1<r<s\leq\infty$, and the first-order fundamental group of $L(\mathbb F_r)$ is $\{1\}$ for all $1<r<\infty$.

This is the proposed analogue for first-order elementary equivalence of the free group factor alternative. The source presents it as conjectural and does not state that either alternative has been established.