  In a previous paper of the authors, we showed that for any polynomials
$P_1,\dots,P_k \in \Z[\mathbf{m}]$ with $P_1(0)=\dots=P_k(0)$ and any subset
$A$ of the primes in $[N] = \{1,\dots,N\}$ of relative density at least
$\delta>0$, one can find a "polynomial progression" $a+P_1(r),\dots,a+P_k(r)$
in $A$ with $0 < |r| \leq N^{o(1)}$, if $N$ is sufficiently large depending on
$k,P_1,\dots,P_k$ and $\delta$. In this paper we shorten the size of this
progression to $0 < |r| \leq \log^L N$, where $L$ depends on $k,P_1,\dots,P_k$
and $\delta$. In the linear case $P_i = (i-1)\mathbf{m}$, we can take $L$
independent of $\delta$. The main new ingredient is the use of the
densification method of Conlon, Fox, and Zhao to avoid having to directly
correlate the enveloping sieve with dual functions of unbounded functions.
