A conjecture believed to be true is the Iwaniec conjecture. I have been studying it recently, and it states that the $p-$norm of the Beurling-Ahlfors transform is $p^*-1$, where $p^*=\max(p,p')$ and $p,p'$ are Hölder conjugate pairs. This can be written as the inequality \begin{equation*} \int_{\mathbb{C}}|\overline{\partial}f|^p\leq (p^*-1)^p\int_{\mathbb{C}}|\partial f|^p. \end{equation*} Note that the case $p=2$ is true (we have a satisfied Hölder conjugate pair). The problem is still open, but is believed to be true for various reasons. The analysis of quasiconformal mappings (via Astala's distortion theorem) implies that the $p-$norm of the Beurling-Ahlfors transform is $(p^*-1)$. The conjecture has been completely settled for so-called "stretch" functions, ie, functions $f:\mathbb{C}\to\mathbb{C}$ of the form $f(z)=g(r)e^{i\theta}$ where $z=re^{i\theta},$ and $g$ is a nonnegative locally Lipschitz function on $(0,\infty)$ such that $g(0)\equiv g(0+)=0$ and $\lim_{r\to\infty}g(r)=0$. There are various estimates of the norm that are approaching the desired bound, and include topics such as Littlewood-Paley inequalities, martingales, and Riesz-Thorin interpolation. All these factors lead people to believe the conjecture to be true since we are approaching $p^*-1$, but it is still open.
The full Fourier restriction problem is believed to be true. This problem seeks inequalities of the form \begin{equation*} ||\widehat{f}||_{L^q(\mathbb{S}^{n-1})}\leq C_{p,q,n}||f||_{L^p(\mathbb{R}^n)} \end{equation*} for a constant $C$ depending on $p,q$ and $n$, and $\mathbb{S}^{n-1}$ is the unit sphere in $\mathbb{R}^n$. There has been progress towards this result suggesting it is true for any $p,q,n$ with famous progress made for the case $q=2$ known as the Tomas-Stein restriction theorem.