I am currently working on a project where one studies a smooth bordered compact $3$-manifold $M$ with some properly embedded essential surface(s) $S \subset M$. More precisely, I am interested in the relative diffeomorphism group $Diff(M, \partial M \cup S)$ and its variation $Diff_D(M, \partial M \cup S)$, where the diffeomorphisms in the second group are required to have all their derivatives at $\partial M \cup S$ agree with those of the identity. My question is the following: Is the inclusion $$ Diff_D(M, \partial M \cup S) \longrightarrow Diff(M, \partial M \cup S) $$ a (weak) homotopy equivalence ? It is mentioned in Alexander Kupers' book on diffeomorphism groups (available at http://people.math.harvard.edu/~kupers/teaching/272x/book.pdf) that the inclusion $$ Diff_D(M, \partial M) \longrightarrow Diff(M, \partial M) $$ is one, but I have not found a reference for the case above.
Any insight is appreciated !
For derivatives up to finite order, this is Corollaire 3 on page 336 of Cerf's Topologie de certains espaces de plongements. The same arguments work for derivatives up to all orders.