(abridged) We present a near-infrared (NIR) photometric variability study of the candidate protoplanet, TMR-1C, located at a separation of about 10" (~1000 AU) from the Class I protobinary TMR-1AB in the Taurus molecular cloud. Our campaign was conducted between October, 2011, and January, 2012. We were able to obtain 44 epochs of observations in each of the H and Ks filters. Based on the final accuracy of our observations, we do not find any strong evidence of short-term NIR variability at amplitudes of >0.15-0.2 mag for TMR-1C or TMR-1AB. Our present observations, however, have reconfirmed the large-amplitude long-term variations in the NIR emission for TMR-1C, which were earlier observed between 1998 and 2002, and have also shown that no particular correlation exists between the brightness and the color changes. TMR-1C became brighter in the H-band by ~1.8 mag between 1998 and 2002, and then fainter again by ~0.7 mag between 2002 and 2011. In contrast, it has persistently become brighter in the Ks-band in the period between 1998 and 2011. The (H-Ks) color for TMR-1C shows large variations, from a red value of 1.3+/-0.07 and 1.6+/-0.05 mag in 1998 and 2000, to a much bluer color of -0.1+/-0.5 mag in 2002, and then again a red color of 1.1+/-0.08 mag in 2011. The observed variability from 1998 to 2011 suggests that TMR-1C becomes fainter when it gets redder, as expected from variable extinction, while the brightening observed in the Ks-band could be due to physical variations in its inner disk structure. The NIR colors for TMR-1C obtained using the high precision photometry from 1998, 2000, and 2011 observations are similar to the protostars in Taurus, suggesting that it could be a faint dusty Class I source. Our study has also revealed two new variable sources in the vicinity of TMR-1AB, which show long-term variations of ~1-2 mag in the NIR colors between 2002 and 2011.

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