Tests of conjectures on multiple Watson values

I define multiple Watson values (MWVs) as iterated integrals, on the interval $x\in[0,1]$, of the 6 differential forms $A=d\log(x)$, $B=-d\log(1-x)$, $T=-d\log(1-z_1x)$, $U=-d\log(1-z_2x)$, $V=-d\log(1-z_3x)$ and $W=-d\log(1-z_4x)$, where $z_1=γ^2$, $z_2=γ/(1+γ)$, $z_3=γ^2/(1-γ)$ and $z_4=γ=2\sin(π/14)$ solves the cubic $(1-γ^2)(1-γ)=γ$. Following a suggestion by Pierre Deligne, I conjecture that the dimension of the space of ${\mathbb Z}$-linearly independent MWVs of weight $w$ is the number $D_w$ generated by $1/(1-2x-x^2-x^3)=1+\sum_{w>0}D_w x^w$. This agrees with 6639 integer relation searches, of dimensions up to $D_5+1=85$, performed at 2000-digit precision, for $w<6$.

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