This is pretty arguable but for Equation (11) perhaps I'd have expected use of $p$ (the observed value of the random variable) rather than $P$ as follows
$$
\mathbb{P}(S_L < S < SR | P = p) = \int{S_L}^{S_R} f_p(y - p) \text{d}y = F_P(S_R - p) - F_P(S_L - p).
$$
Especially because it's a lowercase $p$ in Equation (10).
Also small note but perhaps could use the $\mathbb{P}$ for probability and reserve $P$ for the primary event time random variable? May not cause any confusion though.
This is pretty arguable but for Equation (11) perhaps I'd have expected use of $p$ (the observed value of the random variable) rather than $P$ as follows
$$ \mathbb{P}(S_L < S < SR | P = p) = \int{S_L}^{S_R} f_p(y - p) \text{d}y = F_P(S_R - p) - F_P(S_L - p). $$
Especially because it's a lowercase $p$ in Equation (10).
Also small note but perhaps could use the $\mathbb{P}$ for probability and reserve $P$ for the primary event time random variable? May not cause any confusion though.