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We continue to build support for the proposal to use HII galaxies (HIIGx) andgiant extragalactic HII regions (GEHR) as standard candles to construct theHubble diagram at redshifts beyond the current reach of Type Ia supernovae.Using a sample of 25 high-redshift HIIGx, 107 local HIIGx, and 24 GEHR, weconfirm that the correlation between the emission-line luminosity andionized-gas velocity dispersion is a viable luminosity indicator, and use it totest and compare the standard model $\Lambda$CDM and the $R_{\rm h}=ct$Universe by optimizing the parameters in each cosmology using a maximization ofthe likelihood function. For the flat $\Lambda$CDM model, the best fit isobtained with $\Omega_{\rm m}= 0.40_{-0.09}^{+0.09}$. However, statisticaltools, such as the Akaike (AIC), Kullback (KIC) and Bayes (BIC) InformationCriteria favor $R_{\rm h}=ct$ over the standard model with a likelihood of$\approx 94.8\%-98.8\%$ versus only $\approx 1.2\%-5.2\%$. For $w$CDM (theversion of $\Lambda$CDM with a dark-energy equation of state $w_{\rm de}\equivp_{\rm de}/\rho_{\rm de}$ rather than $w_{\rm de}=w_{\Lambda}=-1$), astatistically acceptable fit is realized with $\Omega_{\rmm}=0.22_{-0.14}^{+0.16}$ and $w_{\rm de}= -0.51_{-0.25}^{+0.15}$ which,however, are not fully consistent with their concordance values. In this case,$w$CDM has two more free parameters than $R_{\rm h}=ct$, and is penalized moreheavily by these criteria. We find that $R_{\rm h}=ct$ is strongly favored over$w$CDM with a likelihood of $\approx 92.9\%-99.6\%$ versus only $0.4\%-7.1\%$.The current HIIGx sample is already large enough for the BIC to rule out$\Lambda$CDM/$w$CDM in favor of $R_{\rm h}=ct$ at a confidence levelapproaching $3\sigma$.

The HII Galaxy Hubble Diagram Strongly Favors $R_{\rm h}=ct$ over $\Lambda$CDM