Number of rectangles to cover a circle

It is possible to cover the circle by $11$ rectangles.

We can construct the $11$ rectangles by following procedure.

• Center the circle of radius $9$ at origin.
• Start covering the circle with rectangle $C_0 = [5,9] \times [-4,4]$ (the red one).
• Rotate $C_0$ with respect to origin for angles $\frac{2k}{7} k \pi $ for $k = 1,\ldots, 6$. This gives us $6$ new rectangles $C_1, C_2, \ldots, C_6$ (the gray ones).
• Making copies of $C_0$, $C_2$ and $C_5$ and shift them inwards for a radial displacement $4$. This give us $3$ rectangles $C'_0$, $C'_2$ and $C'_5$ (the green ones). To make the covering work, one need to shift $C'_2$ and $C'_5$ a little bit tangentially too.
• What's remain on the circle can be covered by the rectangle $[-7,1] \times [-2,2]$ (the blue one).

According to Erich's packing center, the current best known covering of circle by $18$ unit squares has radius $r \approx 2.116$. Since $2.116 \le 2.25 = \frac{9}{4}$, this means there is no known covering of our circle with $9$ rectangles. This leaves us with the question whether we can reduce the number of rectangles from $11$ to $10$.