I just find the following counter-example. Suppose $A,B,C$ are discrete variables. $A,B$ can each take two values while $C$ can take three values. The joint distribution $P(A,B,C)$ is:
\begin{array}{cccc} A & B & C & P(A,B,C) \\ 1 & 1 & 1 & 0.1/3 \\ 1 & 1 & 2 & 0.25/3 \\ 1 & 1 & 3 & 0.25/3 \\ 1 & 2 & 1 & 0.4/3 \\ 1 & 2 & 2 & 0.25/3 \\ 1 & 2 & 3 & 0.25/3 \\ 2 & 1 & 1 & 0.4/3 \\ 2 & 1 & 2 & 0.25/3 \\ 2 & 1 & 3 & 0.25/3 \\ 2 & 2 & 1 & 0.1/3 \\ 2 & 2 & 2 & 0.25/3 \\ 2 & 2 & 3 & 0.25/3 \\ \end{array}
So the marginal distribution $P(A,B)$ is: \begin{array}{ccc} A & B & P(A,B) \\ 1 & 1 & 0.2 \\ 1 & 2 & 0.3 \\ 2 & 1 & 0.3 \\ 2 & 2 & 0.2 \\ \end{array}
The marginal distributions $P(A), P(B)$ and $P(C)$ are uniform.
So we can compute that: \begin{align} d(P1,P3) &= 0.2 \\ d(P2,P3) &= 0.8/3 \end{align}\begin{align} d(P1,P3) &= 0.1 \\ d(P2,P3) &= 0.4/3 \end{align}