Holomorphic mass equidistribution conjecture for Hecke eigencuspforms

Let $\mathbbm{H}$ be the upper half plane with hyperbolic measure $d\mu z=y^{-2}dxdy$, let $\Gamma=\operatorname{SL}_2(\mathbbm{Z})$, and set $X=\Gamma\backslash\mathbbm{H}$. Let $\mathcal{L}(X)$ be the Hilbert space of square-integrable automorphic functions on $X$, with inner product

$$\langle f,g\rangle=\int_X f(z)\overline{g(z)}\,d\mu z.$$

Choose any sequence of holomorphic Hecke eigencuspforms $\{f_k\}$ normalized by

$$\int_X y^k|f_k(z)|^2\,d\mu z=1,$$

and define $F_k(z)=y^{k/2}f_k(z)$ for $z=x+iy$. For a fixed smooth and bounded function $\phi$ on $X$, the holomorphic mass equidistribution conjecture asserts that

$$\langle \phi F_k,F_k\rangle=\frac{1}{\operatorname{Vol}(X)}\langle\phi,1\rangle+o(1)\qquad\text{as }k\longrightarrow\infty.$$

This is a holomorphic analogue of the more general Quantum Unique Ergodicity conjecture for arithmetic surfaces; the stated equidistribution result was subsequently proved, resolving the conjecture.