Does antisymmetricity of $\preceq$ fail in general C*-algebras?

Let $n\geq 3$ and consider the Cuntz algebra $\mathcal {O}_n$, which is generated by isometries $\{S_1,\ldots , S_n\}$, satisfying $\sum_{i=1}^n S_iS_i^*=1$. Working within $M_2( \mathcal {O}_n)$ we have $$ \pmatrix{S_1^* & 0 \cr S_2^* & 0}\pmatrix{S_1 & S_2 \cr 0 & 0} = \pmatrix{S_1^*S_1 &S_1^*S_2 \cr S_2^*S_1 & S_2^*S_2} = \pmatrix{1 & 0 \cr 0 & 1}, $$ while $$ \pmatrix{S_1 & S2 \cr 0 & 0} \pmatrix{S_1^* & 0 \cr S_2^* & 0} = \pmatrix{S_1S_1^* + S_2S_2^* & 0 \cr 0 & 0} \leq \pmatrix{1 & 0 \cr 0 & 0}. $$ So $$ \pmatrix{1 & 0 \cr 0 & 1} \preceq \pmatrix{1 & 0 \cr 0 & 0} $$ and it is clear that $$ \pmatrix{1 & 0 \cr 0 & 0}\preceq \pmatrix{1 & 0 \cr 0 & 1}. $$

However these are not equivalent projections because their class in $K_0(\mathcal {O}_n) = \mathbb {Z}_{n-1}$ differ. Precisely, the $K_0$ class of $\pmatrix{1 & 0 \cr 0 & 0}$ is 1, while the $K_0$ class of $\pmatrix{1 & 0 \cr 0 & 1}$ is 2, and $1\neq 2$ in $\mathbb {Z}_{n-1}$ for $n\geq 3$.