I am fairly new to this, but my attempt below involves using a fourth system R to purify.
$$S(A,B,C)+S(B) \le S(A,B)+S(B,C),$$ introducing a system R which purifies ABC, we get $$S(A,B,R)+S(B) \le S(A,B)+S(B,R).$$
$$S(C)+S(B) \le S(A,B)+S(A,C)\\ \Longrightarrow S(A|C)+S(A|B) \ge0 \\\Longrightarrow -S(A|C)-S(A|B) \le 0$$
$$S(A:C)-S(A)+S(A:B)-S(A) \le 0 \to S(A:C)+S(A:B) \le 2S(A)$$
I'm not sure if my reasoning here is correct. Is is valid to do what I did by subbing in the system R? I am still getting used to proving things in this manner. S is the Von Neumann Entropy, and the individual systems are d-dimensional quantum systems.