cosmology - Conformal time or cosmological time?

Let us assume a flat FRW metric

$$ds^2 = -dt^2 + a(t)^2(dx^2+dy^2+dz^2)$$

Imagine a lightbeam traveling in the x-direction. It travels on a null geodesic with $ds=0$ so that its path obeys the relation

$$dt = a(t) dx$$

Let us assume that $a=1$ at the present cosmological time.

Thus presently in an interval of cosmological time $dt$ light travels a proper distance of $dx$.

Now imagine a future time at which $a=2$.

Thus in the future in the same interval of cosmological time $dt$ a light beam will travel a proper distance of $2dx$.

Is this correct?

It seems that if an observer uses cosmological time as his time measure then the local speed of light will change with the age of the Universe.

However if he uses conformal time $\tau$ such that an interval of conformal time $d\tau$ is given by

$$d\tau = \frac{dt}{a}$$

then while the proper distance the light travels increases in proportion to $a$ the number of time intervals will also be proportional to $a$ (because their size $d\tau$ is inversely proportional to $a$).

Thus the local speed of light will remain constant if one uses conformal time rather than cosmological time (as it should).