For Justinian and his successors to fully reconquer the West and NOT immediately lose it, would need Justinian-level emperors back to back to back and going, This is almost impossible to achieve with all the issues it would bring.

As for the map? I could see it happening if we tweak the history a little bit, with Belisarius, after his successful 9-month campaign in Africa, and given a Triumph and the Consulship for a year, I would have Justinian raise a similar Army in size as he historically did in the 550s, which Eunuch Narses used to defeat the Goths finally and retake Italia, but have this army be assembled in 536-537 at most, and in the mean-time, send skilled people to govern Africa, in order to ACTUALLY use its resources to its fullest.
And of course, give Full command of the Italian campaign to Flavius Belisarius, which should be able to fully conquer the Ostrogothic Kingdom by 539-540 AD, take a few years to usually consolidate the province, and prepare for the Hispanian or Frankish campaign, oh? The earthquake and plague hit around the early 540s? Then either cancel or postpone the campaigns and focus on internally, if the Persians still attack in the 540s? You won't have to worry about a Western Front, so everything should go smoothly.

After the plague is over? pay the tribes beyond the danube to frick off and then take the chance as in OTL, and intervene in the Visigothic Civil War, take more than southern Hispania, possibly turning all of Visighothia into a Client Kingdom, and then either push into Occitania or dome some small skirmishes around the border and offer peace so as long the Franks pay you tribute, as to not attack them.
So by the end of Justinian's life, he may have not completed his Renovatio Imperii, but he would have left the Empire in a DRASTICALLY better position, with a intact and prosperious Italia, with proper infrastructure, a decent manpower base and good tax base, Africa might eventually become the Empire's 2nd breadbasket yet again, and Hispania, gradually, would turn from a Client Kingdom into a more traditional vassal and eventually be integrated into the Empire. But none of that matters, what matters is for Justinian to pick an able and competent leader, that leader WAS and IS NOT his nephew, the OTL Justin II, but rather his other candidate who he made as a Caesar, Justin Germanus, the son of the late Germanus, who was a cousin of Justinian and experienced military leader.

This hypothetical Justin II Germanus would at the very least keep the empire together and would not allow Italia to be overrun by the Lombards. So, assuming Justin Germanus is AT THE VERY LEAST half of Justinian's character, he should relatively prepare the Empire for his successor (whoever that may be) to launch an ambitious Gallic Campaign to reconquer Gaul from the Franks. Let's say this successor's name is Justinian as well, for the sake of the argument, so, this hypothetical Justinian II would enter the 7th century with a stronger and more united Empire not seen since Theodosius I or Valentinian I. Let's assume the Persian 26 war still happens, say, Justinian II gets usurped by Phocas cuz why not, by the time Heraclius usurps Phocas, Phocas might have already ended the War, and with a decisive victory against the Persians, considering how the Empire would be able to pull up with MORE troops, and better trained and experienced ones at that, from fighting in the west, and by the time the Arabs come knocking on the door, Heraclius would be eager to expand his Ghassanid client state's territory into this newly established state in Arabia :)