Is there a conjectured gap between semiprimes?

There is a conjectured gap between primes in form of Cramer's conjecture. Using this we have $p_1\leq p_0+c(\log p_0)^2$ for consecutive primes $p_0$ and $p_1$.

Then we have $s_0=p_0p_1\leq p_0p_2=s_1$ and $p_2p_0-p_1p_0\geq c(\log^2p_0)p_0$. So is the gap between consecutive semiprimes at least $\sqrt{s_0}$.

Could there always or never be a prime smaller than $s_0$ and larger than $s_1$ whose product falls in the interval $[s_0, s_0+\sqrt{s_0}]$?

In general is gap between product of $m$-primes at least $s^{\frac{m-1}m}\log^2s$?