$\newcommand{\ep}{\varepsilon}$This conjecture is not true in general.
Indeed, suppose the "convex" part of your conjecture is true. Then (letting $x:=\phi$, $t:=\theta_1$, and $\theta_2\downarrow\theta_1=t$) we see that for any strictly increasing convex smooth function $g$ and all $x$ and $t$ in $(0,\pi/2)$ we would have $h_2(g;x,t):=\partial_x\partial_t\,\ln(g(\cos^2(x-t))-g(\cos^2(x+t))\ge0$. (Note that for all $x$ and $t$ in $(0,\pi/2)$ we have $\cos^2(x-t)-\cos^2(x+t)=\sin2x\,\sin2t>0$, so that $h_2(g;x,t)$ is well defined.)
For real $\ep>0$ and real $u$, let $u_{+;\ep}:=\frac12(u+\sqrt{\ep^2+u^2})$, an "$\ep$-smoothed" version of $u_+:=\max(0,u)$. For $c$ and $c_*$ in $[0,\infty)$, let $g_{c_*,\ep}(c):=(c-c_*)_{+;\ep}$.
Then the function $g_{c_*,\ep}$ is strictly increasing, convex, and smooth on $[0,\infty)$. However, $h_2(g;x,t)=-44051.358\ldots\not\ge0$ if $g=g_{c_*,\ep}$, $c_*=\frac12$, $\ep=\frac1{1000}$, $x=\frac{39}{100}$, and $t=\frac{118}{100}$. So, the "convex" part of your conjecture is not true in general.
Suppose now the "concave" part of your conjecture is true. Then for any strictly increasing concave smooth function $g$ and all $x$ and $t$ in $(0,\pi/2)$ we would have $h_2(g;x,t)\le0$.
For $c$ and $c_*$ in $[0,\infty)$, let $G_{c_*,\ep}(c):=c-\sqrt{\ep^2+(c-c_*)^2}$.
Then the function $G_{c_*,\ep}$ is strictly increasing, concave, and smooth on $[0,\infty)$. However, $h_2(G;x,t)=32614.565\ldots\not\le0$ if $G=G_{c_*,\ep}$, $c_*=\frac12$, $\ep=\frac1{1000}$, and $x=\frac{39}{100}=t$. So, the "concave" part of your conjecture is not true in general either. $\quad\Box$