Non-rationality conjecture for Fano threefolds of type X_(1,1,1,1)

Let $X$ be a Fano threefold over a field ${\mathsf{k}}$ of type $\mathsf{X}_{(1,1,1,1)}$, and let $\uprho(X)$ denote its geometric Picard rank. Assume

$$\uprho(X)=1.$$

Non-rationality conjecture. Then $X$ is never ${\mathsf{k}}$-rational.

This would complete the rationality criteria for the six types of Fano threefolds considered in the paper. The source states that a partial result is known, but does not provide enough information here to determine whether the conjecture has been resolved.