  The electron-electron Coulomb interaction in Dirac-Weyl semimetals harbours a
novel paradigm of correlation effects that hybridizes diverse realms of
solid-state physics with their relativistic counterpart. Driving spontaneous
mass acquisition, the excitonic condensate of strongly-interacting massless
Dirac fermions is one such example whose exact nature remains debated. Here, by
focussing on the two-dimensional tilted Dirac cones in the organic salt
$\alpha$-(BEDT-TTF)$_2$I$_3$, we show that the excitonic instability is
controlled by a small chemicalpotential shift and an in-plane magnetic field.
In combined analyses based on renormalization-group approaches and ladder
approximation, we demonstrate that the nuclear relaxation rate is an excellent
probe of excitonic-spin fluctuations in an extended parameter region.
Comparative nuclear magnetic resonance (NMR) experiments show good agreements
with this result, jointly revealing the importance of intervalley nesting
between field-induced, spin-split Fermi pockets of opposite charge polarities.
Our work provides an accurate framework to search for excitonic instability of
strongly-interacting massless fermions.
