Two synchronised clocks accelerated perpendicular to Earth's gravitational field

Question

If I have two synchronized (at the station while waiting for me) clocks on a train, one at front and the other one at back, and I start to accelerate the train perpendicular to Earth's gravitational field then do those clocks go out-of-sync? Say, I accelerate at rate 0.1 m/s^2 till the speed equals 20 m/s and the distance between the clocks is 100 m.

Are those clocks out-of-sync when I travel comfortably after the acceleration?

Edit: Using gravitational time dilation for solving the problem sounds odd when I'm not accelerating along a gravitational field.

Edit 2: If I accelerated perpendicularly to a gravitational field I should have, according to the equivalence principle, experienced out-of-sync for my clocks. Does there exist any experimental evidence for the phenomenon? Theory isn't a proof.

possible duplicate of Which clock is the fastest inside an accelerating body? — John Rennie, Sep 11 '15 at 16:33
Hi Kimmo. In the duplicate I've suggested it's an accelerating rocket not an accelerating train, but the argument is the same. — John Rennie, Sep 11 '15 at 16:34
@JohnRennie So it won't make any difference if I accelerate perpendicular to a gravitational field? Is there any experimental evidence about the phenomenon where the acceleration occurred perpendicular to a gravitational field? I mean using gravitational time dilation sounds a bit odd if I'm not accelerating along a gravitational field. — Kimmo Rouvari, Sep 11 '15 at 17:18
Correct. The fact that we can use the ideas of gravitational time dilation when describing the effect of acceleration is called the equivalence principle and it is at the core of general relativity. — John Rennie, Sep 11 '15 at 19:23

Two synchronised clocks accelerated perpendicular to Earth's gravitational field

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