Assume:<br/>
-. Test current = 500mA<br/>
-. LED headlight takes below 500mA to lit, and drops 10V.<br/>

Approximated voltage and time correlation is: dV = 1/C * I * dt<br/>
To hold voltage below 10V for 10 seconds: C = (500mA * 10sec) / 10V = 0.5F<br/>

So, combination of super capacitors with 0.5F/12V may do the job, for this hypothetical case. Meantime, I thought, LED headlights use inline converter(voltage boost/current regulation). If so, the story will be different.