  Bell inequalities are mathematical constructs that demarcate the boundary
between quantum and classical physics. A new class of multiplicative Bell
inequalities originating from a volume maximization game (based on products of
correlators within bipartite systems) has been recently proposed. For these new
Bell parameters, it is relatively easy to find the classical and quantum, i.e.
Tsirelson, limits. Here, we experimentally test the Tsirelson bounds of these
inequalities using polarisation-entangled photons for different number of
measurements ($n$), each party can perform. For $n=2, 3, 4$, we report the
experimental violation of local hidden variable theories. In addition, we
experimentally compare the results with the parameters obtained from a fully
deterministic strategy, and observe the conjectured nature of the ratio.
Finally, utilizing the principle of "relativistic independence" encapsulating
the locality of uncertainty relations, we theoretically derive and
experimentally test new richer bounds for both the multiplicative and the
additive Bell parameters for $n=2$. Our findings strengthen the correspondence
between local and nonlocal correlations, and may pave the way for empirical
tests of quantum mechanical bounds with inefficient detection systems.
