Anyone?

We're essentially asked (in a more fun way) if the area of the circle = the area of the square, so let's try to answer that.

S1::

Since the quadrilateral has four equal sides, it is either a square or a rhombus. If the side is s, then the area is either s² or unknown, if we have a rhombus.

Naming the radius of the circle r, we've also got r√π = s, or r = s/√π. So we know the area of the circle is π * (s/√π)², or s².

So if the quadrilateral is a circle, the murals have equal area, but if it's only a rhombus they may not: INSUFFICIENT.

S2::

Since the quadrilateral has equal diagonals, it could be a square, a rectangle, or an isosceles trapezoid. We don't know which, so we can't find the area of the quadrilateral: INSUFFICIENT.

S1 + S2::

Together, we know from S1 that we've got either a square or a rhombus, and that each side = r√π. From S2, we know that the diagonal of the quadrilateral = r * √2π. If our quadrilateral is a square, we'd apply the Pythagorean Theorem: Side² + Side² = Diagonal². Here, that gives us

(r√π)² + (r√π)² = (r√2π)²

or

2r²π = 2r²π

Success! So it's a square, and we're set: SUFFICIENT.