The authors present the even-$N$ moments $N \leq 20$ of the fourth-order (N$^3$LO) contribution $P_{\text{gq}}^{(3)}(x)$ to the quark-to-gluon splitting function in perturbative QCD. These moments, obtained by analytically computing off-shell operator matrix elements for a general gauge group, agree with all known results, particularly with the moments $N \leq 10$ derived from structure functions in deep-inelastic scattering. Using these new moments and endpoint constraints, they construct approximations for $P_{\text{gq}}^{(3)}(x)$ that improve upon those obtained from the lowest five even moments. The remaining uncertainties of this function are now practically irrelevant at momentum fractions $x > 0.1$. The resulting errors of the convolution of $P_{\text{gq}}$ at N$^3$LO with a typical quark distribution are small at $x \gtrsim 10^{-3}$ and exceed 1\% only at $x \lesssim 10^{-4}$ for a strong coupling $\alpha_s = 0.2$. The present results for $P_{\text{gq}}^{(3)}(x)$ should thus be sufficient for most collider-physics applications.The authors present the even-$N$ moments $N \leq 20$ of the fourth-order (N$^3$LO) contribution $P_{\text{gq}}^{(3)}(x)$ to the quark-to-gluon splitting function in perturbative QCD. These moments, obtained by analytically computing off-shell operator matrix elements for a general gauge group, agree with all known results, particularly with the moments $N \leq 10$ derived from structure functions in deep-inelastic scattering. Using these new moments and endpoint constraints, they construct approximations for $P_{\text{gq}}^{(3)}(x)$ that improve upon those obtained from the lowest five even moments. The remaining uncertainties of this function are now practically irrelevant at momentum fractions $x > 0.1$. The resulting errors of the convolution of $P_{\text{gq}}$ at N$^3$LO with a typical quark distribution are small at $x \gtrsim 10^{-3}$ and exceed 1\% only at $x \lesssim 10^{-4}$ for a strong coupling $\alpha_s = 0.2$. The present results for $P_{\text{gq}}^{(3)}(x)$ should thus be sufficient for most collider-physics applications.