  We present initial results of an ongoing survey with the Karl G. Jansky Very
Large Array targeting the CO($J$ = 1-0) transition in a sample of 30
submillimeter-selected, dusty star-forming galaxies at $z =$ 2-5 with existing
mid--$J$ CO detections from ALMA and NOEMA, of which 17 have been fully
observed. We detect CO(1-0) emission in 11 targets, along with three tentative
($\sim$1.5-2$\sigma$) detections; three galaxies are undetected. Our results
yield total molecular gas masses of 6-23$\times$10$^{10}$
($\alpha_\mathrm{CO}$/1) M$_\odot$, with gas mass fractions,
$f_\mathrm{gas}$=$M_\mathrm{mol}$/($M_*$+$M_\mathrm{mol}$), of 0.1-0.8 and a
median depletion time of (140$\pm$70) Myr. We find median CO excitation ratios
of $r_{31}$ = 0.75$\pm$0.39 and $r_{41}$ = 0.63$\pm$0.44, with a significant
scatter. We find no significant correlation between the excitation ratio and a
number of key parameters such as redshift, CO(1-0) line width or
$\Sigma_\mathrm{SFR}$. We only find a tentative positive correlation between
$r_{41}$ and the star-forming efficiency, but we are limited by our small
sample size. Finally, we compare our results to predictions from the SHARK
semi-analytical model, finding a good agreement between the molecular gas
masses, depletion times and gas fractions of our sources and their SHARK
counterparts. Our results highlight the heterogeneous nature of the most
massive star-forming galaxies at high-redshift, and the importance of CO(1--0)
observations to robustly constrain their total molecular gas content and ISM
properties.
