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) &= 0.1 \\
d(P2,P3) &= 0.4/3
\end{align}