I believe it is the general opinion, at least among those working in diophantine approximations, that the extreme case of Salem numbers (the question of the title) would be just as difficult as the full Lehmer conjecture. It is not a coincidence that the smallest known Mahler measures are realized by Salem numbers, and I have not heard of any improvement on the general Dobrowolski bound $\log{M(P)} > \big(\frac{9}{4} - o(1) \big) \Big( \frac{\log{\log{d}}}{\log{d}} \Big)^3$ under restricting to the Salem case. As far as I know, not even an improvement on the constant $9/4$, which is due to Loubotin, is possible under assuming that $P$ is the minimum polynomial of a Salem number $\lambda$ (whereby $M(P) = \lambda$).

I believe it is the general opinion, at least among those working in diophantine approximations, that the extreme case of Salem numbers (the question of the title) would be just as difficult as the full Lehmer conjecture. It is not a coincidence that the smallest known Mahler measures are realized by Salem numbers, and I have not heard of any improvement on the general Dobrowolski bound $\log{M(P)} > \big(\frac{9}{4} - o(1) \big) \Big( \frac{\log{\log{d}}}{\log{d}} \Big)^3$ under restricting to the Salem case.

[The constant $9/4$, due to Loubotin in 1983, is apparently the best that Dobrowolski's method can produce. Voutier has shown that the inequality holds without exception with the constant $1/4$.]

I believe it is the general opinion, at least among those working in diophantine approximations, that the extreme case of Salem numbers (the question of the title) would be just as difficult as the full Lehmer conjecture. It is not a coincidence that the smallest known Mahler measures are realized by Salem numbers, and I have not heard of any improvement on the general Dobrowolski bound $\log{M(P)} > \frac{1}{4} \Big( \frac{\log{\log{d}}}{\log{d}} \Big)^3$ under restricting to the Salem case.

I believe it is the general opinion, at least among those working in diophantine approximations, that the extreme case of Salem numbers (the question of the title) would be just as difficult as the full Lehmer conjecture. It is not a coincidence that the smallest known Mahler measures are realized by Salem numbers, and I have not heard of any improvement on the general Dobrowolski bound $\log{M(P)} > \big(\frac{9}{4} - o(1) \big) \Big( \frac{\log{\log{d}}}{\log{d}} \Big)^3$ under restricting to the Salem case. As far as I know, not even an improvement on the constant $9/4$, which is due to Loubotin, is possible under assuming that $P$ is the minimum polynomial of a Salem number $\lambda$ (whereby $M(P) = \lambda$).