Equivalence of Padmanabhan’s light-cone thermodynamic gravity and Modified Measure Theories via an independent conformal factor

Determine whether Padmanabhan’s mesoscopic, light-cone-based variational framework for gravity and Modified Measure Theories (MMT) are equivalent formulations once the metric’s conformal factor is treated as an independent degree of freedom; specifically, establish whether the two approaches are "two sides of the same coin" under this condition.

Background

The paper links Padmanabhan’s statistical/thermodynamic approach to gravity—where null vectors on the light cone serve as configuration variables—to Modified Measure Theories (MMT), which achieve invariance to additive constants in the Lagrangian by replacing the metric measure with a metric-independent measure.

Within this context, the authors explicitly mention a conjecture that these two approaches may be essentially equivalent if the conformal factor of the metric is treated as an independent variable, but they do not provide a rigorous argument, highlighting the need for a formal equivalence proof.