  The Gamow-Teller strength distribution from ${}^{88}$Sr was extracted from a
$(t,{}^{3}\text{He}+\gamma)$ experiment at 115 MeV/$u$ to constrain estimates
for the electron-capture rates on nuclei around $N=50$, between and including
$^{78}$Ni and $^{88}$Sr, which are important for the late evolution of
core-collapse supernovae. The observed strength below an excitation energy of 8
MeV was consistent with zero and below 10 MeV amounted to $0.1\pm0.05$. Except
for a very-weak transition that could come from the 2.231-MeV $1^{+}$ state, no
$\gamma$ lines that could be associated with the decay of known $1^{+}$ states
were identified. The derived electron-capture rate from the measured strength
distribution is more than an order of magnitude smaller than rates based on the
single-state approximation presently used in astrophysical simulations for most
nuclei near $N=50$. Rates based on shell-model and quasiparticle random-phase
approximation calculations that account for Pauli blocking and
core-polarization effects provide better estimates than the single-state
approximation, although a relatively strong transition to the first $1^{+}$
state in $^{88}$Rb is not observed in the data. Pauli unblocking effects due to
high stellar temperatures could partially counter the low electron-capture
rates. The new data serves as a zero-temperature benchmark for constraining
models used to estimate such effects.
