  We point out that the symplectic structure, written in terms of the
Sen-Ashtekar-Immirzi-Barbero variables, of a spacetime admitting an isolated
horizon as the inner boundary, involves a positive constant parameter, say
$\sigma$, if $\gamma\neq\pm i$, where $\gamma$ is the Barbero-Immirzi
parameter. The parameter $\sigma$ represents the rescaling freedom that
characterizes the equivalence class of null generators of the isolated horizon.
We reiterate the fact that the laws of mechanics associated with the isolated
horizon does not depend on the choice of $\sigma$ and, in particular, while one
uses the value of standard surface gravity as input, that does not fix $\sigma$
to a particular value. This fact contradicts the claims made in certain parts
of the concerned literature that we duly refer to. We do the calculations by
taking Schwarzschild metric as an example so that the contradiction with the
referred literature, where similar approaches were adopted, becomes apparent.
The contribution to the symplectic structure that comes from the isolated
horizon, diverges for $\sigma^2=(1+\gamma^2)^{-1}$, implying that the rescaling
symmetry of the isolated horizon is violated for any real $\gamma$. Since the
quantum theory of $SU(2)$ isolated horizon in the LQG framework exists only for
real values of $\gamma$, it is founded on this flawed classical setup.
Nevertheless, if the flaw is ignored, then two different viewpoints persist in
the literature for entropy calculation. We highlight the main features of those
approaches and point out why one is logically viable and the other is not.
