co.combinatorics - Why are multinomial coefficients with same entropy equal? (usually)

Suppose $p_1,ldots,p_d$ and $q_1,ldots,q_d$ are positive real numbers such that

$$p_1+cdots+p_d=q_1+cdots+q_d=n$$

and

$$p_1 log p_1+cdots+p_dlog p_d=q_1 log q_1+cdots+q_d log q_d$$

Then the following seems to hold

$$frac{n!}{p_1!cdots p_d!}=frac{n!}{q_1!cdots q_d!}$$

why?

Edit: JBL correctly notices that it doesn't always hold. I just didn't go far enough.
Still, it's surprising to me that it holds so frequently.

If we put a black disk at x,y if equality seems to hold (in machine precision) for x=n,y=d, and positive integer coefficients, it'll look like this

Red circle is JBL's example. Blue circle is n=18,d=3 which fails for (12,3,3) and (9,8,1).

docheck[n_, d_] := (
   coefs = IntegerPartitions[n, {d}, Range[1, n]];
   entropy[x_] := N[Total[# Log[#] & /@ x]];
   groupedCoefs = GatherBy[coefs, entropy];
   allEqual[list_] := And @@ (First[list] == # & /@ list);
   multinomials = Apply[Multinomial, groupedCoefs, {2}];
   And @@ (allEqual /@ multinomials)
   );
vals = Table[docheck[#, d] & /@ Range[1, 30], {d, 1, 20}];
Graphics[Table[Disk[{n, d}, If[vals[[d, n]], .45, .1]], {d, 1, Length[vals]}, {n, 1, 30}]]

Edit:
Updated version that does exact checking and allows coefficients with 0 components. Still only one example of failure for d=3.

docheck[n_, d_] := (coefs = IntegerPartitions[n, {d}, Range[0, n]];
   entropy[x_] := Exp[Total[If[# == 0, 0, # Log[#]] & /@ x]];
   groupedCoefs = GatherBy[coefs, entropy];
   allEqual[list_] := And @@ (First[list] == # & /@ list);
   multinomials = Apply[Multinomial, groupedCoefs, {2}];
   And @@ (allEqual /@ multinomials));
maxn = 30; maxd = 20;
vals = Table[docheck[#, d] & /@ Range[1, maxn], {d, 1, maxd}];
Graphics[Table[Disk[{n, d}, If[vals[[d, n]], .45, .1]], {d, 1, Length[vals]}, {n, 1, maxn}]]

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