= Solution
Write $a_n=|\mathcal P_n|$ for the number of labelled $r$-<uniform hypergraphs> with the <hereditary hypergraph property>, so $d_n=a_n^{1/\binom nr}$ for $n\geq r$. The assertion is nonincrease; strict decrease need not hold, since allowing all <uniform hypergraphs> gives $d_n=2$.
If $a_n=0$, heredity forces $a_{n+1}=0$. If $a_{n+1}=0$, the desired inequality is immediate. Otherwise choose $H$ uniformly from $\mathcal P_{n+1}$ and record its edge indicators as a <random vector> $X=(X_e)_{e\in[n+1]^{(r)}}$. Its <information entropy>, with any fixed logarithm base, is $H(X)=\log a_{n+1}$. Deleting a vertex $v$ gives the restriction $X^{(v)}$ to the edges avoiding $v$. By heredity and relabelling, this restriction has at most $a_n$ possible values, so $H(X^{(v)})\leq\log a_n$.
Each edge coordinate occurs in exactly $n+1-r$ of these restrictions. For completeness, order all the edge coordinates. The <chain rule for information entropy> expands a restriction entropy as a sum of conditional coordinate entropies. The predecessors retained in a restriction form a subset of all predecessors, and <conditioning reduces entropy>; hence each restriction term is at least the corresponding full-vector term. Summing over the restrictions counts each full-vector term $n+1-r$ times. This proves the needed <Shearer inequality>:
$$
(n+1-r)\log a_{n+1}\leq\sum_{v=1}^{n+1}H(X^{(v)})\leq(n+1)\log a_n.
$$
Since $\binom{n+1}r=\frac{n+1}{n+1-r}\binom nr$, division gives $\log d_{n+1}\leq\log d_n$. Therefore the <normalized speed of a hereditary hypergraph property> satisfies
$$
\boxed{d_{n+1}\leq d_n\qquad(n\geq r).}
$$
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