Assume $d=\min \{a, b, c, d\}$, and let $x=a+d$, then
$$\frac{a}{\sqrt{a+b+c}}+\frac{d}{\sqrt{d+a+b}} \leqslant \frac{a}{\sqrt{a+b+d}}+\frac{d}{\sqrt{d+a+b}}=\frac{x}{\sqrt{x+b}}$$

It is also clear that
$$\frac{b}{\sqrt{b+c+d}} \leqslant \frac{b}{\sqrt{b+c}}$$

Thus, it suffices to prove
$$\frac{x}{\sqrt{x+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+d+a}} \leqslant \frac{5}{4} \sqrt{a+b+c+d}$$

The above inequality is the Jack Garfunkel inequality (with variables $x, b, c$).
The equality holds if and only if $\frac{a}{3}=\frac{b}{1}=\frac{c}{0}=\frac{d}{0}$ and its cyclic permutations.