A single-phase room-temperature multiferroic with strong magnetoelectric coupling
In plain words
BiFeO3 is magnetic and ferroelectric at room temperature but has almost no net magnetization, while materials with strong coupling between the two orders work only at low temperature. No single material yet combines both at room temperature.
Precise statement
Identify a single-phase compound with ferroelectric polarization above about $3e3\ \mathrm{statC}/\mathrm{cm}^{2}$ and net magnetization above $10\ \mathrm{emu}/\mathrm{cm}^{3}$ at $T = 300\ \mathrm{K}$, in which an electric field deterministically reverses (or rotates by $90\ \text{degrees}$) the magnetization. An answer is a material with measured P, $M$ and demonstrated electric-field reversal of M at $300\ \mathrm{K}$ without a ferromagnetic overlayer.
What would settle it
Synthesis and measurement of a single-phase compound showing 300 K polarization and magnetization hysteresis plus electric-field-driven magnetization reversal imaged by XMCD-PEEM or magnetic force microscopy.
Status in the literature
Deterministic switching at $300\,\mathrm{K}$ so far uses BiFeO3 coupled to a ferromagnetic layer (Heron et al., Nature 2014).