Can We Reach Chromatic 5 Without Mosers Spindles?

Is there a unit-distance 5-chromatic graph (in the plane) without Mosers Spindle? I was sure there was. This is a natural question that naturally came to my mind while posing relevant open problems. After all, Aubrey de Grey used a dense in spindles construction because it worked, not because it would not work otherwise. Without Mosers Spindles, we achieve much gain in flexibility of embedding, and thus, possibly, getting a smaller graph. And so, I posed the following problem: