For purposes of this question "terminal symplectic variety" means a normal variety which is symplectic (in the usual sense of symplectic singularities) and whose singular locus has codimension $\geq 4$ (the equivalence of this with the usual definition of terminal is a theorem of Namikawa).
Symplectic varieties has lots of nice properties; For example, they are always Cohen-Macaulay, which is a condition about niceness with respect to depth. So, one can hope for others.
Is a terminal symplectic variety necessarily $S_4$?
I should warn any potential answerers that my understanding of the $S_4$ property is very poor (it's an exceptionally tough thing to Google, since it's not even the dominant use of that term in mathematics). I hesitate to even give a definition for fear of messing it up; I believe it means that every ideal sheaf of codimension $\leq 3$ has depth equal to its codimension.