(Dis)prove $\frac{4(3\sqrt2-4)}{a+b+c+d}+\sum\limits_{cyc}\frac1{\sqrt a}\geq3\sqrt2$ for positive $a$, $b$, $c$, $d$ with $\sum_{cyc}\frac1{1+a}=2$

An open problem from Art of Problem Solving (AoPS):

If $a,b,c,d$ are positive real numbers such that$$\frac{1}{1+a}+\frac 1{1+b}+\frac 1{1+c}+\frac 1{1+d}=2,$$then prove or disprove$$\frac1{\sqrt a}+\frac1{\sqrt b}+\frac1{\sqrt c}+\frac1{\sqrt d}+\frac{4\left(3\sqrt2-4\right)}{a+b+c+d}\geq3\sqrt2$$

(V. Cîrtoaje and L. Giugiuc, 2022) Link to AoPS and some progress to the problem: https://artofproblemsolving.com/community/u399206h2794888p24621900

Some Progress

by dragonheart6Some thoughts (the method of Lagrange Multiplier):

At minimum, $(a, b, c, d)$ satisfies$$F(a) = F(b) = F(c) = F(d) > 0 \qquad (1)$$where$$F(u) := \frac{\lambda}{(1 + u)^2} - \frac{1}{2u^{3/2}}.$$($\lambda$ is the Lagrange Multiplier.)According to (1), try to prove that $|\{a, b, c, d\}| \le 2$ where $|\cdot|$ is the cardinality of a set.