  Background: Recent developments in {\it ab initio} nuclear theory demonstrate
promising results in medium- to heavy-mass nuclei. A particular challenge for
many of the many-body methodologies, however, is an accurate treatment of the
electric-quadrupole, $E2$, strength associated with collectivity. Purpose: The
valence-space in-medium similarity renormalization group (VS-IMSRG) is a
particularly powerful method for accessing medium- and high-mass nuclei but has
been found to underpredict $E2$ strengths. The purpose of this work is to
evaluate the isospin dependence of this underprediction. Methods: We perform a
systematic comparison of valence-space in-medium similarity renormalization
group (VS-IMSRG) calculations with available literature. We make use of
isoscalar and isovector contributions to the $E2$ matrix elements to assess
isoscalar and isovector contributions to the missing strength. Results: It is
found that the $E2$ strength is consistent throughout
$T_z=\left|\frac{1}{2}\right|$, $T_z=\left|1\right|$,
$T_z=\left|\frac{3}{2}\right|$ and $T_z=2$ pairs within the $sd$-shell.
Furthermore, no isovector contribution to the deficiency is identified.
Conclusions: A comparison with toy-models and coupled-cluster calculations is
used to discuss potential origins of the missing strength, which arises from
missing many-particle, many-hole excitations out of the model space. The
absence of any significant isovector contribution to the missing $E2$ strength
indicates that the $E2$ strength discrepancy, and therefore any correction, is
largely independent of the isospin of the nuclei in question.
