Conjectures on radii, densities, and symmetries of maximally circlable circles

For a nonnegative integer $n$, an MC-circle is a largest circle enclosing exactly $n$ lattice points in its interior, and $R_n$ is its radius; an integer is MC if it is maximally circlable and non-MC otherwise. A lattice circle is a circle with at least three lattice points on its circumference. For $n\ne6$, let $\rho_n$ be the radius of the largest $n$-enclosing lattice circle when one exists, and set $\rho_n=0$ otherwise; a strong MC number is an MC number immediately followed by a non-MC number. MC-circle distribution conjectures. The following assertions hold: for MC numbers $k<\ell$, $R_k<R_\ell$; every non-MC $n\ne6$ has a largest $n$-enclosing lattice circle; the values in $\{\rho_n\}_{n\ge0}$ are pairwise distinct; there are arbitrarily long consecutive sequences of MC numbers and of non-MC numbers; the density of MC numbers is greater than $80\%$, while that of non-MC numbers is positive and less than $20\%$; strong MC numbers are infinite and have positive density among MC numbers; every MC-circle of a strong MC number has at least six lattice points on its circumference; the stated majority, minority, and percentage claims about mirror symmetries and axes of symmetry in items (10)--(19) hold exactly as listed in the source. These claims are extrapolated from computations for $n<1100$ and remain unresolved in the paper.