Conjecture on the gap between the smallest Salem numbers

Let $\theta_n^{-1}$ denote the relevant Perron-number sequence, and let a Salem number be a real algebraic integer greater than $1$ whose conjugates satisfy the usual Salem-number modulus conditions. Salem-number gap conjecture. There is no Salem number in the interval

$$(\theta_{31}^{-1},\theta_{12}^{-1})=(1.08544\ldots,1.17295\ldots).$$

This would improve the known lower bound $\theta_{31}^{-1}$ for Salem numbers and would place the next possible Salem number above $\theta_{12}^{-1}$. The source presents the assertion as a conjecture; no resolution evidence is supplied.