The proof of the result that every convergent net in a uniform space is Cauchy, employs symmetry of the uniform space. A quasiuniform space lacks that symmetry. Is it possible then to find a convergent net in a quasiuniform space which is not Cauchy?

$\begingroup$ Posted also at math.SE: Convergent net which is not Cauchy. $\endgroup$ – Martin Sleziak Oct 1 '17 at 6:12
There are several definitions of Cauchy filters on a quasiuniform space $(X,\mathcal U)$ [K]. For instance, a filter $\mathcal F$ on $(X,\mathcal U)$ is called
a left $K$Cauchy (resp. right $K$Cauchy) filter, if for each $U\in\mathcal U$ there is $F\in\mathcal F$ such that $U(x)\in \mathcal F$ (resp. $U^{1}(x)\in \mathcal F $) whenever $x\in F$;
a $\mathcal U^*$Cauchy filter, if for each $U\in\mathcal U$ there is $F\in\mathcal F$ such that $F\times F\subset U$;
a $D$Cauchy filter, if there exists a cofilter $\mathcal G$ on $X$ (that is, for each $U\in\mathcal U$ there are $F\in\mathcal F$ and $G\in\mathcal G$ such that $G\times F\subset U$;
a PS (that is, PervinSieber)Cauchy filter, if for each $U\in\mathcal U$ there is $x\in X$ such that $U(x)\in\mathcal F$;
a weakly Cauchy filter, if for each $U\in\mathcal U$, $\bigcap_{F\in\mathcal F} U^{1}(F)\ne\varnothing$.
Convergent filters are not necessarily “eachCauchy”. For instance, in general convergent filters are not left $K$Cauchy: [K] e.g. a regular quasimetric space in which each convergent sequence has a left $K$Cauchy subsequence is metrizable (see [KMRV, Proposition 4]).
References
[K] H.P.A. Künzi, Quasiuniform Spaces in the Year 2001 (in Recent Progress in General Topology II, pages 313344; link to preprint).
[KMRV] H.P.A. Künzi, M. Mršević, I. L. Reilly, and M. K. Vamanamurthy, Convergence, precompactness and symmetry in quasiuniform spaces, Math. Japonica 38 (1993), 239253.